Let X n be a bounded sequence of integers. Show that X n has a subsequence that

Question

Let Xn be a bounded sequence of integers. Show that Xn has a subsequence that is eventuallyconstant. So, i know that by the Bolzano-Weierstrass Thm for sequencesthat since Xn is bounded that it must have a convergentsubsequnce, but i dont really know what to do from there...any helpwould be greatly appreciated Let Xn be a bounded sequence of integers. Show that Xn has a subsequence that is eventuallyconstant. So, i know that by the Bolzano-Weierstrass Thm for sequencesthat since Xn is bounded that it must have a convergentsubsequnce, but i dont really know what to do from there...any helpwould be greatly appreciated

Explanation / Answer

If L is the limit of the subsequence, then L must be aninteger. If it weren't, then for E < (distance from L tothe nearest integer), there would be no elements of the wholesequence within a distance E of L. Thus the subsequence couldnot converge to L. If you let E = 1/2, and let L be the limit of the subsequence, thenfor some N, you have | x_n - L | < 1/2 for every n

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